Mathematics HL and further mathematics HL formula booklet

Diploma Programme

Mathematics HL and further mathematics HL

formula booklet

For use during the course and in the examinations First examinations 2014

Edited in 2015 (version 2)

? International Baccalaureate Organization 2012

5048

Contents

Prior learning

2

Core

3

Topic 1: Algebra

3

Topic 2: Functions and equations

4

Topic 3: Circular functions and trigonometry

4

Topic 4: Vectors

5

Topic 5: Statistics and probability

6

Topic 6: Calculus

8

Options

10

Topic 7: Statistics and probability

10

Further mathematics HL topic 3

Topic 8: Sets, relations and groups

11

Further mathematics HL topic 4

Topic 9: Calculus

11

Further mathematics HL topic 5

Topic 10: Discrete mathematics

12

Further mathematics HL topic 6

Formulae for distributions

13

Topics 5.6, 5.7, 7.1, further mathematics HL topic 3.1

Discrete distributions

13

Continuous distributions

13

Further mathematics

14

Topic 1: Linear algebra

14

Mathematics HL and further mathematics formula booklet

1

Formulae

Prior learning

Area of a parallelogram

A= b ? h , where b is the base, h is the height

Area of a triangle

=A 1 (b ? h) , where b is the base, h is the height 2

Area of a trapezium

=A 1 (a + b) h , where a and b are the parallel sides, h is the height 2

Area of a circle

A = r2 , where r is the radius

Circumference of a circle

C= 2r , where r is the radius

Volume of a pyramid=V 1 (area of base ? vertical height) 3

Volume of a cuboid

V =l ? w ? h , where l is the length, w is the width, h is the height

Volume of a cylinder

V = r2h , where r is the radius, h is the height

Area of the curved surface of A= 2rh , where r is the radius, h is the height

a cylinder

Volume of a sphere

V= 4 r3 , where r is the radius 3

Volume of a cone

V= 1 r2h , where r is the radius, h is the height 3

Distance between two

points (x1, y1) and (x2 , y2 )

d = (x1 - x2 )2 + ( y1 - y2 )2

Coordinates of the midpoint of a line segment with endpoints

(x1, y1) and (x2 , y2 )

x1

+ 2

x2

, y1

+ 2

y2

Solutions of a quadratic equation

The solutions of ax2 + bx + c =0 are x = -b ? b2 - 4ac 2a

Mathematics HL and further mathematics formula booklet

2

Core

Topic 1: Algebra

1.1 The nth term of an

arithmetic sequence

un = u1 + (n -1) d

The sum of n terms of an

arithmetic sequence

Sn=

n 2

(

2u1

+

(n

-

1)

d

)=

n 2

(u1

+

u n

)

The nth term of a

geometric sequence

un = u1r n-1

The sum of n terms of a finite geometric sequenc= e Sn

u= 1(rn -1) r -1

u1(1 - rn ) , r 1 1- r

The sum of an infinite geometric sequence

S

=

u1 1- r

,

r

0, b > 0, a 1

a x = exln a

loga a x= x= aloga x

logb

a

=

logc logc

a b

1.3 Combinations

n r

=

r

n! !(n -

r)!

Permutations

n Pr

=

n! (n - r)!

Binomial theorem 1.5 Complex numbers 1.7 De Moivre's theorem

(a + b)n

=

an

+

n 1

a n -1b

++

n r

an-rbr

+ + bn

z =a + ib =r (cos + isin ) =rei =r cis

[r(cos + i sin )]n =rn (cos n + i sin n ) =rnein =rn cis n

Mathematics HL and further mathematics formula booklet

3

Topic 2: Functions and equations

2.5 Axis of symmetry of the graph of a quadratic function

f (x) =ax2 + bx + c axis of symmetry x =- b 2a

2.6 Discriminant

= b2 - 4ac

Topic 3: Circular functions and trigonometry

3.1 Length of an arc

l = r , where is the angle measured in radians, r is the radius

Area of a sector

A = 1 r2 , where is the angle measured in radians, r is the 2

radius

3.2 Identities

tan = sin cos

sec = 1 cos

cosec = 1 sin

Pythagorean identities

cos2 + sin2 = 1 1 + tan2 = sec2 1 + cot2 = csc2

3.3 Compound angle identities sin ( A= ? B) sin Acos B ? cos Asin B cos ( A ? B) = cos Acos B sin Asin B tan ( A ? B) =tan A ? tan B 1 tan A tan B

Double angle identities

sin 2 = 2sin cos cos 2 = cos2 - sin2 = 2cos2 -1 = 1 - 2sin2 tan 2 = 2 tan

1 - tan2

Mathematics HL and further mathematics formula booklet

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